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An interesting trigonometric integral

Once while solving problems from Romanian Mathematical Olympiads I encountered a mathematical gem. The problem I found was asking to calcualte the following integral \[I=\int\frac{\cos x}{a\sin x+b\cos x}\,dx.\]

What do I find especially beautiful about this problem? It rewards the desire to examine a bit more than is asked initially!

Solution. Let's try to evaluate one more integral \[J=\int{\frac{\sin x}{a\sin x+b\cos x}}\,dx.\]

Can you guess the next step?

Yes, we will take advantage of the fact that the sum of integrals is the integral of sum. In other words: \[aJ+bI=a\int{\frac{\sin x}{a\sin x+b\cos x}}\,dx+b\int{\frac{\cos x}{a\sin x+b\cos x}}\,dx\\=\int{\frac{a\sin x+b\cos x}{a\sin x+b\cos x}}\,dx=x+C.\]

On other hand we can easily calculate integrals of the form \(\int{\frac{f^\prime(x)}{f(x)}}\,dx\). The derivative of \( (a\sin x+b\cos x)\) is equal to \( (a\cos x-b\sin x)\), which can also be expressed as a linear combination of our two integrals, i.e. \[aI-bJ=\int{\frac{a\cos x-b\sin x}{a\sin x+b\cos x}}\,dx=\ln{|a\sin x+b\cos x|}+C.\]

Now we simply have to solve a system of linear equations. We obtain \[I=\frac{a\ln{|a\sin x+b\cos x|}+bx}{a^2+b^2}+C.\] and \[J=\frac{ax-\ln{|a\sin x+b\cos x|}}{a^2+b^2}+C.\]

Has anyone of you seen problems with the similar ideas? How often while solving a problem you investigate possible variations of the conditions?

Note by Nicolae Sapoval
3 years, 3 months ago

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Try solving this the same way: \(\Large I = \int \frac {a \sin x + b \cos x }{c \sin x + d \cos x} \,dx \) Pi Han Goh · 3 years, 3 months ago

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@Pi Han Goh

  1. Lets split the integral \(I\) into the sum of two integrals: \[I=a\int\frac{\sin x}{c\sin x+d\cos x}\,dx+b\int\frac{\cos x}{c\sin x+d\cos x}\,dx=aJ+bK.\]
  2. Now we reduced our problem to the previous one, because integrals \(J,K\) are exactly the same as the one mentioned in the post.
Nicolae Sapoval · 3 years, 3 months ago

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@Nicolae Sapoval Smiley_Face.gif Pi Han Goh · 3 years, 3 months ago

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This looks very useful, thanks for sharing Nicolae. :) Pranav Arora · 3 years, 3 months ago

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Thanks for sharing !! Purvam Modi · 3 years, 3 months ago

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Thanks for sharing Rajath Krishna R · 3 years, 2 months ago

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